# DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra

arXiv: https://arxiv.org/abs/2609.03801 · HTML: https://mlsanity.com/paper/2609.03801/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 11 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Built on by (most cited first, 1 of 1)

- [Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates](https://mlsanity.com/paper/2609.17899/) (2026)

## Abstract (arXiv)

We present the discrete--complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel f(p,n,k)=pk(1-p)n-k and its inverse-integer level sets. The binary level 2-n selects p=1/2 as the unique real split-independent anchor. Imposing the additional continuation rule 1-z=overline z gives the conjugation-symmetric line z=1/2+iu. On the central normalized branch, the inverse-level split kappa=k/n also has real part 1/2. The quadratic coordinate Q(z)=1/4+u2 identifies complementary conjugate points, has minimum 1/4, and has an explicit two-sign inverse; restricting its values to integers defines an arithmetic quantization. For supplied positive critical-line zero ordinates γk, the induced level has the exact decomposition [ Lk=frac14+γk2=Nk+δk,qquad δk=widetilde B1!left(γk2+frac34right), ] where Nk=lfloor Lk+1/2rfloor and widetilde B1(x)=x-1/2. The integer--phase pair (Nk,e2π iδk) retains the level and the positive ordinate. Prime-exponent coordinates describe the factorization of the integer component, while conditional Fourier moments formulate an open question about the residual phases. The finite numerical diagnostics are descriptive. A distinct complex exponent s gives an inverse-level representation of the supplied term m-s; direct substitution cancels the auxiliary coordinates. These terms assemble into the classical Dirichlet series and Euler product for Re s>1. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.